Question: An archaeologist discovers a triangular stone slab with side lengths 13 cm, 14 cm, and 15 cm. What is the radius of the inscribed circle?
["Title: How to Calculate the Radius of the Inscribed Circle Around a Triangle with Sides 13 cm, 14 cm, and 15 cm", "---", "Introduction:\nMathematics and archaeology often intersect in fascinating ways, especially when ancient artifacts reveal harmonic geometric properties. Recently, an archaeologist uncovered an intriguing triangular stone slab with side lengths of 13 cm, 14 cm, and 15 cm. Beyond its historical value, this perfectly preserved triangle provides an excellent real-world example to calculate the radius of the inscribed circle—an essential concept in geometry. In this article, we explore how to determine the inradius of such a triangle, transforming ancient stone into a compelling lesson in applied mathematics.", "---", "### Understanding the Inscribed Circle (Inradius)", "The inscribed circle, also known as the incircle, is the largest circle that fits perfectly inside a triangle, tangent to all three sides. The radius of this circle (r) is known as the inradius. It plays a vital role in geometry, particularly in formulas involving area, semiperimeter, and triangle proportions.", "---", "### Step-by-Step Calculation: Finding the Inradius of a 13–14–15 Triangle", "Given triangle sides:\na = 13 cm,\nb = 14 cm,\nc = 15 cm.", "#### Step 1: Compute the Semi-Perimeter\nThe semi-perimeter (s) is half the perimeter of the triangle:\n[\ns = \frac{a + b + c}{2} = \frac{13 + 14 + 15}{2} = \frac{42}{2} = 21 \ ext{ cm}\n]", "#### Step 2: Calculate the Area Using Heron’s Formula\nHeron’s formula gives the area (A) of a triangle with sides a, b, c:\n[\nA = \sqrt{s(s - a)(s - b)(s - c)}\n]\nSubstituting the values:\n[\nA = \sqrt{21(21 - 13)(21 - 14)(21 - 15)} = \sqrt{21 \ imes 8 \ imes 7 \ imes 6}\n]\nFirst simplify inside the square root:\n[\n21 \ imes 8 = 168,\quad 7 \ imes 6 = 42,\quad 168 \ imes 42 = 7056\n]\nSo:\n[\nA = \sqrt{7056} = 84 \ ext{ cm}^2\n]", "#### Step 3: Use the Formula for the Inradius\nThe formula relating area, semi-perimeter, and inradius is:\n[\nA = r \cdot s \quad \Rightarrow \quad r = \frac{A}{s}\n]\nSubstitute known values:\n[\nr = \frac{84}{21} = 4 \ ext{ cm}\n]", "---", "### Conclusion: A Practical Insight from Ancient Stone\nThe inscribed circle inscribed in the triangular stone slab with sides 13 cm, 14 cm, and 15 cm has a radius of 4 centimeters. This geometric feature not only highlights the precision of ancient craftsmanship but also offers a tangible example for students and enthusiasts to understand the relationship between a triangle’s side lengths and its internal circle.", "If you're exploring geometry through real artifacts like this, remember: ancient shapes whisper timeless mathematical truths.", "---", "### Key Takeaways\n- The triangular stone slab has side lengths 13 cm, 14 cm, and 15 cm.\n- Its semi-perimeter is 21 cm.\n- Its area is 84 cm² using Heron’s formula.\n- The radius of the inscribed circle (inradius) is 4 cm.", "Cymour archaeology isn’t just about uncovering history—it’s about unveiling the elegant math embedded within.", "---", "Keywords for SEO:\ntriangle inscribed circle radius, inradius calculation, 13-14-15 triangle, Heron’s formula, archaeological geometry, circle in a triangle, geometric properties of ancient stone, radius of inscribed circle, math in archaeology", "Meta Description:\nDiscover how to calculate the radius of the inscribed circle in a triangle with sides 13 cm, 14 cm, and 15 cm. Learn step-by-step geometry math behind this ancient stone slab's precise design. Ideal for students and math enthusiasts."]
